How Uniform is the Uniform Distribution on Permutations?

Tech Report Number

649

For large q, does the (discrete) uniform distribution on the set of q! permutations of the vector (1, 2, . . . , q) closely approximate the (continuous) uniform distribution on the (q2)-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a (q 1)-dimensional convex polyhedron. Surprisingly to me, the answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, (1,2,...,q) is not the most favorable configuration for approximate spherical uniformity of permutations. Unlike the permutations of (1, 2, . . . , q), the normalized surface area of the largest empty spherical cap among the permutations of the most favorable configuration approaches 0 as q ! 1. Several open questions are posed.